Issue84-Variograms

as the “range” (a). If at a distance nearly

equal to zero, i.e., h 0, the variogram valueis greater than zero, this value is known asthe “nugget-effect” (C0). The total-sill of thevariogram (S) is C+C0. Often C is also treatedequal to the sill of the variogram model fittedto the experimental variograms and the nuggeteffect (C0). Both C0 and the sill (S) characterizethe random aspect of the data, where as therange (a) and C characterize the structuralaspect of the deposit.

Assumptions

We should remember that the practice ofgeostatistics requires two assumptions1, 2 of 1)second order stationarity (i.e., mean andcovariance of the variable is invariant undertranslation), and 2) intrinsic hypothesis (i.e., themean and covariance of the increments Z(x+h)-Z(x) exist and are independent of the point x). Insimple language this mean that a geologicallyhomogenous domain can be divided into anumber of domains with the same mean andco-variances, which means that a variogrammodel fitted to a particular set of data,represents any subset of data from the samepopulation. So, a variogram represents ageneral spatial statistical model of a data-set.What does that mean geologically or in miningterms? In other words, does a variogram makeany sense to the geology? A geologist/geostatistician/ mining engineer should alwaystry to answer this question by relating the

ISSUE NO. 84 — May 2007

Introduction

The three major functions used ingeostatistics for describing the spatialcorrelation of observations are thecorrelogram, the covariance, and the semi-variogram. The last is also more simply calledthe variogram. The variogram is the keyfunction in geostatistics as it is used to fit amodel of the spatial correlation of the data.

Modeling variograms for mineral resourceestimation is a common practice. Variogrammodels are used in kriging estimationprocedures, conditional-simulation of amineral deposit, and also to infer maximumdistances of spatial autocorrelation (ranges)which can further be used in construction ofsearch parameters for different interpolationtechniques. Typically, different sets ofvariograms are made for one variable indifferent rock-types. For example, iron (Fe)concentration in limonitic saprolite and gabbroicsaprolite are modeled separately.

Variogram Parameters

A variogram represents both structural andrandom aspects of the data underconsideration. The range of a variogramrepresents the structural part of the variogrammodel. The variogram values increase withincreases in the distance of separation until itreaches the maximum (C) at a distance known

Copyright 2007 by Pincock, Allen and Holt, a division of Runge Inc. All Rights Reserved.

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C A L E N D A R○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○

Basics of Variogram Analysis

Consultants for Mining and Financial Solutions

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e-mail: [email protected] 1 Isaaks, E, H., and Srivastava R. M., 1989, An Introduction to Applied Geostatistics, Oxford University Press2 Armstrong, M., 1998, Basic Linear Geostatistics, Springer

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major direction produces the best resultbecause it enables using a simple ellipsoidfor the variogram analysis. This ellipsoidcan be used to define search parameterstoo. A variogram map can also be madefor multivariate cases, where a cross-covariance map provides a sense ofmultivariate anisotropy. More detailsabout variogram maps can be found inany major geostatistics textbook.

Variogram Calculation

Before calculating a variogram, severaldata analyses should be done to ensurethat the data comes from a single(statistical) population, one singlehomogenous geological/geochemicalunit without any major structuraldisplacements. For an example, a singleunfolded coal seam bound by otherlithological layers (e.g., sandstone/shale)and any planar structural elements(faults) should be considered as ahomogenous geological unit. Mixedpopulations of data should be split intosubsets with unique population

variogram information to “real world”geologic conditions seen in the field, andin the case of operating mines, actualproduction data. This is because avariogram model represents the spatialstructure of the data set; so a variogrammodel for a stockwork-vein gold depositshould be different from a typical low-grade epithermal gold deposit. Theranges of the latter are expected to belonger along any lateral-direction thanthe former case. In the case of layereddeposits like coal, the variogram of thedata (such as carbon content in coal)with periodic variation with depth, mayshow a cyclic-pattern of the sill (hole-effect)4 model along a vertical direction,which is not common in the case of amassive sulfide type mineral deposit suchas a porphyry-copper deposit.

Omnidirectional vs. DirectionalVariograms

Depending on the complexity of thegeology, and the statistical and spatialcharacteristics of the data under

consideration, a more complexvariogram may be required. Anomnidirectional variogram may besufficient for kriging data with auniform distribution in a lithologic unitwith little or no structural discontinuity,such as Ca-content in a limestonedeposit. A geologically complex mineraldeposit may require a set of anisotropicvariograms with more than one“structure” in each direction.

To assess the degree of anisotropy andmajor-direction of spatial continuity(along which the variogram range isexpected to be largest among alldirections), a variogram map is a veryuseful tool (Figure 1). As seen in Figure1, the variable, Fe, seems to be spatiallyautocorrelated for a longer distancealong the NE-SW orientation; henceNE-SW should be considered as themajor orientation for calculation of thedirectional variogram. Even though onecan find more than one minor directionfor any given variable, typically theminor direction perpendicular to the ○

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N270

N120

N300

N150

N330

N180N2

10

N30

N240

N60

U

V

N/A

253.1267246.4130239.6994232.9857226.2720219.5583212.8447206.1310199.4173192.7037185.9900179.2763172.5626165.8490159.1353152.4216145.7079138.9943132.2806125.5669118.8533112.1396105.4259 98.7122 91.9986 85.2849 78.5712 71.8576 65.1439 58.4302 51.7165 45.0029 38.2892

FIGURE 1An Example of a Variogram Map (Fe content

in a limonitic rock type)

FIGURE 2A Spherical Variogram Model Fitted to the

Experimental-Variogram Points

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parameters because variogram analysisusing mixed populations can producemisleading results. Further, spatialdistribution of the samples should beexamined and care should be taken toreduce the impact of “data clustering.” Apopular way of dealing with the clustereffect is to assign weights to the samplesduring variography. Sometimes drill-holedata are logged at unequal lengthintervals. If variograms are calculated withsamples of various sizes, the variogrammay not represent the underlyingstructure of the variable in consideration.Samples of unequal size may not producea realistic representative variogrambecause a larger sample may not provideinformation of equal weighting as thesmaller samples do. Compositing the drill-hole data at equal length should reducethis effect (support-effect). Ideally, drill-holes of unequal size should be treatedseparately or their thickness should beweighed differently for variographyanalyses.

Variogram Modeling

As discussed earlier, depending on thetype of data under consideration and thestructural and lithological nature of thegeology of the deposit, the variogramanalysis may be as simple as a singleomnidirectional variogram or as complexas multiple structures along each directionin a set of anisotropic variograms. In thelatter case, it is advised to perform thefollowing analyses in order to develop areasonable variogram model.

An important first step of a variogramanalysis is to model and review theomnidirectional variogram to infer the sillof the anisotropic variograms. The total sillof an omnidirectional variogram providesan approximate idea of the total sill ofspecific directional variograms. The totalsill of the directional variogram should be

less than or equal to the sample(population) variance (V). Because abovethe variance line (see Figure 2) thesample behaves as a random variable,the variogram values above the variance(V) should not be included for thevariogram model, hence are not helpfulin modeling.

The slope at the beginning of thevariogram model (at 0 < h < R)represents the structural part of thevariogram. The tangent drawn to thispart of the variogram meets the sill atapproximately 95 percent of the range.This property is helpful in estimatingthe range from the first fewexperimental-variogram points at thebeginning.

The nugget-effect represents the

randomness of the variables at h 0,which is a characteristic of the variable,hence should remain constant in alldirections. A nugget-effect for a variablecan be calculated by using theomnidirectional variogram. But if thedrill-holes are vertical and equal-lengthcomposited drill-hole data are available,the nugget-effect can preferably becalculated by modeling down-the-hole(DTH) vertical variogram. This normallyprovides the closest pairs forinterpretations close to the zerodistance.

The other major aspect of directionalvariogram modeling is examination ofanisotropy of the variable in the deposit.For more details of the anisotropyreaders are referred to Armstrong(1998) 2, Isaaks and Srivastava (1989)1,Wackernegel (2003) 3, Journel andHuigibregts (2004) 4.

When experimental variograms of avariable show different behaviors(different variogram parameters) in

different directions, an anisotropic setof variograms should be fitted to theexperimental variogram. If the sills ofthe variograms change with directionso that they cannot be considered thesame for all directions, a variogrammodel with components having “Zonalanisotropy” can be fitted to theexperimental variograms. A modelingtechnique is proposed by Deraisme5 toaddress this situation. If the sills of thevariograms are comparable but theranges of the variograms change withdirection, a variogram with componentsof “geometric anisotropy” type shouldbe fitted to the experimental variogram.In this case the nugget effect and thesills of the variograms should remainconstant in all directions. It is importantto note that only admissible models,namely spherical, exponential, hole-effect, nugget-effect, gaussian model,etc., should be fitted to theexperimental variograms. These modelsensure the positive definite condition ofthe variograms, which means that thecovariance of the variable must bepositive definite functions2.

Summary

In summary, the ten steps listed belowshould be followed to develop areasonable variogram model for adeposit:

1) Review the geology: One variableper lithologic unit should beanalyzed and modeled at a time.Ore bodies controlled by differentstructural units should also betreated separately for this purpose.Vein type mineralization should betreated separately from thedisseminated type deposit and datain the oxidized zone should beseparated from the unoxidizedzones for variography.

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3 Wackernagel, H., 2003, Multivariate Geostatistics: An Introduction with Applications, Springer4 Journel, A. G., and Huijbregts , C, J., 2004, Mining Geostatistics, The Blackburn Press5 Zonal Anisotropy: how to model the variogram? at www.geovariances.com/IMG/pdf/Zonal_Anisotropy.pdf

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2) Sample size: Review the data set forthe deposit, specifically differentsample length, drill-hole size,channel-sample vs. R.C. drill-holesamples, etc. Samples of differentsizes should be separated intodifferent groups for variogramanalyses or different weights shouldbe assigned to samples of differentsizes to ensure the minimumsupport-effect.

3) Sample distribution: To avoid thecluster effect in areas where samplesare closely spaced, the samplesshould be de-clustered. Samples withan irregular distribution should bevisually analyzed to ensure anapproximately uniform sampledistribution for variography. Low-angle drill-hole samples and channelshould be treated carefully to avoidunrealistic mathematical artifacts.Such samples should be declusteredor should not be included (with otheruniformly distributed samples) duringvariography.

4) Statistical characteristics: Performclassical statistical analysis on the dataset to identify data set issues andmultiple populations. Includecalculating means, ranges, standarddeviations, coefficients of variation,etc., and create cumulative frequencydistribution plots, histograms, andscatter plots of data as necessary togain an understanding of the natureof the element. If required, aGaussian transformation (whichtransforms the original distributionof the variable into a normaldistribution) should be performedbefore variography. The readers are

cautioned that such a transformationcan be complex and will require apost-processing (post kriging or postsimulation) transformation to bringthe data back to its originalcharacteristics.

5) Cleanse the data set if required(e.g., apply a cap and cut assayvalues above the cap or throw outhigh fliers or erroneous lookingdata). Composites of too small alength should be filtered out, as toomany of them may create problemsrelated to the support effect.

6) Generate omnidirectionalexperimental variograms for eachpopulation and identify the sill.

7) Generate down-the-holevariograms for the primary elementin order to identify your nuggetvalue. Sometimes if samples arewell distributed in an equi-dimensional 3-D grid, anomnidirectional variogram can alsobe used for estimating the nugget.

8) Analyze the variogram maps toidentify the anisotropy and majordirection(s) of variogram analyses.

9) Generate multi-directionalvariograms along identifieddirections and model them.

10) Repeat for other elements ofinterest and look for correlationsbetween elements such as gold/silver, gold/arsenic, etc. This maylead to multi-variate cross-variogram analyses for co-krigingor, conditional-simulation.

Concluding Remarks

The application of variogram modelingin designing search and estimationparameters for kriging and othersimulation methodologies are well-known in the mining industry. Goodvariogram analysis can provide anunderstanding of population and spatialinformation important to the productionof a realistic grade or lithologic model.An unrealistic nugget or sill can result inerroneous variances (and errors) ofestimation and hence make it difficult toobtain good deposit models. Also,mistakes made in interpretation ofvariogram ranges often lead tooverestimation of resource tonnes.

An important first step in variographyanalysis is a detailed statistical analysis ofthe data set. The spatial distribution ofsamples should be carefully examined todetermine if the data is adequate forgeostatistical (variogram) analysis. If not,then a statistical analysis should beconducted to prepare for estimationusing interpolation techniques otherthan kriging. A typical reason that datasets are not conducive to variogramanalysis is that there are not enoughsample points of a given variable withina lithological unit, spread uniformly todevelop a good structure.

Acknowledgment:

This month’s article was provided byAbani Samal, Ph.D., Geologist/Geostatistician [email protected] to Mr. Jeffrey Duvall and Mr.Richard Lambert for their insightfulcomments and suggestions.

Issue84-Variograms

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